Regularity of Lie groups

We solve the regularity problem for Milnor’s infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. Specifically, we prove that if $G$ is an infinite dimensional Lie group in Milnor’s sense, then the evolution map is $C^0$-continuous on its domain $\operatorname{iff}$ $G$ is locally $\mu$-convex – This is a continuity condition imposed on the Lie group multiplication that generalizes the triangle inequality for locally convex vector spaces. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete – This is a completeness condition formulated in terms of the Lie group operations that generalizes Mackey completeness as defined for locally convex vector spaces; so that we generalize the well known fact that a locally convex vector space is Mackey complete if each smooth (compactly supported) curve is Riemann integrable. Then, under the presumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve, for $k \subset \mathbb{N}_{\geq 1} \sqcup {\lbrace \mathsf{lip},\infty \rbrace}$, is integrable (contained in the domain of the evolution map) $\operatorname{iff}$ $G$ is Mackey complete and $k$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence consisting of piecewise integrable curves – A similar result is proven for the case $k \equiv 0$; and we provide several mild conditions that ensure that $G$ is $k$-confined for each $k \in \mathbb{N} \sqcup {\lbrace \mathsf{lip},\infty \rbrace}$. We finally discuss the differentiation of parameter-dependent integrals in the standard topological context ($C^k$-topology). In particular, we show that if the evolution map is defined and continuous on $C^k ([0, 1], \mathfrak{g})$ for $k \in \mathbb{N} \sqcup {\lbrace \infty \rbrace}$, then it is smooth thereon:\begin{align}\textrm{For} \: k = 0: & \quad \textit{iff} \quad \textrm{it is differentiable at zero} \\& \quad \textit{iff} \quad \mathfrak{g} \: \textrm{is integral complete.} \\\textrm{For} \: k \in \mathbb{N}_{\geq} \sqcup {\lbrace \infty \rbrace}: & \quad \textit{iff} \quad \textrm{it is differentiable at zero} \\& \quad \textit{iff} \quad \mathfrak{g} \: \textrm{is Mackey complete.}\end{align}This result is obtained by calculating the directional derivatives explicitly – recovering the standard formulas (Duhamel) that hold, e.g., in the Banach (finite dimensional) case.

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Received 24 September 2018

Accepted 10 July 2019

Published 22 July 2022